On structure of uniformly distributed sequences in $[-\frac{1}{2},\frac{1}{2}]$ from the point of view of shyness

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Abstract

In the paper [G.Pantsulaia, On Uniformly Distributed Sequences on [-1/2,1/2], Georg. Inter. J. Sci. Tech.,4(3) (2013), 21--27], it was shown that $\mu$-almost every element of $\mathbf{R}^{\infty}$ is uniformly distributed in $[-\frac{1}{2}, \frac{1}{2}]$, where $\mu$ denotes Yamasaki-Kharazishvili measure in $\mathbf{R}^{\infty}$ for which $\mu([-\frac{1}{2},\frac{1}{2}]^{\infty})=1$. In the present paper the same set is studying from the point of view of shyness and it is demonstrated that it is shy in $\mathbf{R}^{\infty}$. In Solovay model, the set of all real valued sequences uniformly distributed module 1 in $[-\frac{1}{2}, \frac{1}{2}]$ is studied from the point of view of shyness and it is shown that it is the prevalent set in $\mathbf{R}^{\infty}$.